09-240/Classnotes for Tuesday December 1

~In the above gallery, there is a complete copy of notes for the lecture given on December 1st by Professor Natan (in PDF format).

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MAT240 – December 1st

Basic Properties of ${\displaystyle \det :\mathbb {M} _{n\times n}\rightarrow F}$:

(Note that det(EA) = det(E)·det(A) and that det(A) may be written as |A|.)

0. ${\displaystyle \,\!\det(I)=1}$

1. ${\displaystyle \det(E_{i,j}^{1}A)=-\det(A);|E_{i,j}^{1}|=-1}$

Exchanging two rows flips the sign.

2. ${\displaystyle \det(E_{i,c}^{2}A)=c\cdot \det(A);|E_{i,c}^{2}|=c}$

These are "enough"!

3. ${\displaystyle \det(E_{i,j,c}^{3}A)=\det(A);|E_{i,j,c}^{3}|=1}$

Adding a multiple of one row to another does not change the determinant.

The determinant of any matrix can be calculated using the properties above.

Theorem:

If ${\displaystyle {\det }':\mathbb {M} _{n\times n}\rightarrow F}$ satisfies properties 0-3 above, then ${\displaystyle \det '=\det }$

${\displaystyle \det(A)=\det '(A)}$

Philosophical remark: Why not begin our inquiry with the properties above?

We must find an implied need for their use; thus, we must know whether a function ${\displaystyle \det }$ exists first.